Download ON g-CLOSED SETS AND g

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Kyungpool Mathematical JOl1TI1 al
Vol. 33 , No.2 , 20 5- 209 , Oecember 1993
ON g-CLOSED SETS AND g-CONTINUOUS
MAPPINGS
Miguel Caldas Cueva
Introduction. The concept of generalized closed set of a topological
space (breifly g-closed) was introduced by N. Levine ([8] in 1970). These
sets where also considered by W. Dunhan ([5] in 1982) and 야
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H. Maki ([1] in 1991) , defined a new class of mappings called generalized
continuous mappings (briefly g-continuous) which contains the class of
continuous mappings. The present note has as purpose generalize and
improve Theorem 6.3 of N. Levine [8] and to investigate some properties
of g-closed sets and g-continuous mappings. Throughout the present note
X and Y will always denote topological spaces on which no separation
axioms are assumed unless explicity stated. By! X • Y we denote
a mapping not necessarily continuous ! of a topological space X into a
topological space Y. When A is a subset of a topological space , we denote
the closure of A and the interior of A of Cl(A) and Int(A) , respectively.
Definition 1. A subset B of a topological space X is called g-closed in
X [8] if CI(B) c G whenever B c G and G is open in X. A subset A is
called g-open in X if its complement , X - A is g-closed.
Definition 2. A mapping ! : X • Y from a topological space X into a
topological space Y is called g-continuous [1] if the inverse image of every
closed set in Y is g-closed in X. Clearly it is proved that a mapping
!:X • Y is a g-continuous if and only if the inverse image of every open
set in Y is g-open in X.
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Definition 3 . The intersection of all g-closed sets containing as set A
is called the g-closure of A[5] and is denoted by CI*(A). This is , for any
9
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205
206
A c X , CI*(A)
g-closed }
Miguel Caldas Cueva
= n{F : A c
If A
c X,
F E
r}
where
r =
{F : F
c X
and F is
then A 드 Cl*(A) 드 Cl(A).
For the rest of this article we shall assume that C 1“ (A) is g-closed.
Remark 4. A set B is g-closed if and only if Cl*(B) = B.
Definition 5. Let p be a point of X and N be a subset of X. N is called
a g-neighborhood of p in X if there exists a g-open set 0 of X such that
P E 0 c N.
Lemma 6. Let A be a subset of X. Then J p E Cl*(A)
텐
and only if for
any g-neighborhood N p of p in XJ A n N p ::f. 0.
Proof Necessity. Suppose that p E C I*(A). If there exists a g-neighborhood
N of the point p in X such that NnA = 0, then by Definition 5 , there exists
a g-open set Op such that p E Op c N. Therefore , we have Op n A = 0,
so that A c X - Op. Since X - Op is g-closed (Definition 1) , then
CI*(A) c X - Op. As p tf- Cl*(A) which is contrary to the hypothesis
If p tf- Cl ’“ (A) , then by Definition 3 there exists a g-closed set F of X
such that A c F and p tf- F. Therefore , we have p E X - F such that
X - F is a g-open set. Hence X - F is a g-neighborhood of p in X , but
(X - F) n A = 0. This is contrary to the hypothesis.
Reasoning as in [10] we obtain:
Definition 7. Let A be a subset of X. A mapping r : X • A is called a
9 continuous retraction if r is g-continuous and the restriction r / A is the
identity mapping on A.
Theorem 8. Let A be a subset of X and r : X • A be a g-co π tinuous
retraction. lf X is HausdorjJ, then A is a g-closed set of X.
Proof Suppose that A is not g-closed. Then by Remark 4 , there exists a
point x in X such that x E Cl*(A) but x tf- A. It follows that r(x) ::f. x
because r is g-continuous retraction. Since X is Hausdorff, there exists
disjoint open sets U and V in X such that x E U and r(x) E V. Now let
W be an arbitrary g-neighborhood of x. Then W n U is a g-neighborhood
of x. Since x E Cl*(A) , by Lemma 6, we have (WnU)nA 폼 0. Therefore ,
there exists a point y in W n U n A. Since y E A , we have r( ν) = Y E U
and hence r(y) tf- V. This implies that r(W) rt V because y E W. This is
contrary to the g-continuity of r. Consequent1 y, A is a g-closed set of X.
Theorem 9 .
Let {Xdi E I} be
a때
family of topological spaces. lf
On g-closed sets and g-continuous mapping
f: X •
207
IIX‘ is a g-continuous mapping J then Pri 0 f : X • X i is gfor each i E 1J ψhere Pr‘ is the projection of IIX j onto X i .
contiημ ous
Proof. We shall consider a fixed i E 1. Suppose Ui is an arbitrary open set
in X i . Then Pr:;l(Ui ) is open in IIX Since f is g-continuous , we have
by Definition 2, f-1(Pr:;1(Ui )) = (Pri 0 J) -l(Ui ) g-open in X. Therefore
Pri 0 f is g-continuous.
‘.
N. Levine in ([8 ], Theorem 6.3) showed that if f : X • Y is continuous
and closed and if B is a g-open (or g-closed) subset of Y , then f-1(B) is
g-open (or g-closed) in X. The following theorem is a sligh improvement
of this Theorem.
Theorem 10. 1f f : X • Y is g-co η tinuous and closed and if G is a
g-ope η (or g-closed) subset of Y J then f-1( G) is g-ope η (or g-closed) in
X.
Proof. Let G be a g-open set in Y. Let F C f-1 ( G) w here F is closed in X.
Therefore f(F) c G holds. By Theorem 4.2 of [8] and since f(F) is closed
and G is g-open in Y , f(F) c 1nt(G) holds. Hence F C f-1 (I nt(G)).
Since f is g-continuous and 1nt( G) is open in Y ,
F
c
1nT (J -1 (I nt(G)))
c
1nt (J -1(G)) .
Therefore by Theorem 4.2 of [8 ], f-1 (G) is g-open in X. By taking complements , we can show that if G is g-closed in Y , then f-1 (G) is g-closed
in X.
Under g-continuous open mappings , the inverse image of g-open (or
g-closed) sets need not however , be g-open (or g-closed).
Example 1 1. Let X = Y = {a , b, c} , T = {0 , {a }, {c} , {a , c} , X} and a =
{0 , {a} , {a , b} , Y}. Let f: (X ,T) • (Y,σ) be defined by f(a) = f(c) = a
and f( b) = b. Then f is g-continuous and open. However {a , c} is g-closed
in Y but f-1( {a , c}) is not g-closed in X.
By Remark 2 of [1] , the composition mapping of two g-continuous
mappings is not always g-continuous. However , we obtain the following
corollaryas an immediate consequence of Theorem 10.
Corollary 12. Let X , Y and Z be three topological spaces. 1f f : X • Y
is an closed and g-continuous mapping and g : Y • Z is a g-continuo μs
mappz ηgJ then g 0 f : X • Z is g-continμ ous.
208
Miguel Caldas Cueva
We can conclude our work by generalysing the following well known
Theorem which may be found in [3 , Theorem 1. 5 pg.140].
Theorem Å. Let X be an arbitrary topological space , Y be a Haμsdorff
space , and f ,9 : X • Y be continuous. Then {x : f(x) = g(x)} is closed
in X.
Before we prove the generalization of Theorem A , we will need some
preliminary results , analogous to that obtained by D. Jankovic [6] .
Theorem 13. If f : X • Y is g-continuous, ψith Y Hausdorff space , and
F a closed sμ bset of X x Y , the π Prl(F n G (J)) is g-closed in X , where
G (J) = {(x , f(x)) : x E X} denote the graph of f aπ d Prl : X x Y • X
represents the projection.
Proof Let x E C I*(Prl(F n G (J)), where F is a closed subset of X x Y ,
let 0 be an arbitrary open set containing x in X , and let V be an arbitrary
open set containing f(x) in Y. Since f is g-continuous , f-l(V) is g-open
in X. By Corollary 2.7 of [8] 0 n f-l(V) is g-open in X. By Lemma 6
and since x E Cl*(Prl(F n G (J))),
(0 n f-l(V))
n Prl(F n G (J)) =f.
ø.
Let p E (Onf-1(V))nPrl(FnG (J)). This implies that (p , f(p)) E F and
that f(p) E V. Therefore (Ox V) n F =f. and consequently, (x , f(x)) E
CI*(F). Since F is closed (x , f(x)) E FnG (J). Hence x E Prl(FnG (J))
and the result follows of Remark 4.
ø
Corollary 14. If f : X • Y has a closed graph and 9
g-continu01꾀 then {x E X : f(x) = g(x)} is g-closed.
X •
Y is
Proof Since {x E X : f(x) = g(x)} = Prl(G (J) n G(g)) and since G (J)
is a closed subset of X x Y. The result follows from Theorem 13.
A mapping
f :X
•
Y is said to be weakly-continuous [7] if for every
x E X and every open set V containing f( x) in Y there exists an open set
U containing x in X such that f(U) 드 CI(V). Since a weakly-continuous
mapping into a Hausdorff space has a closed graph ([9] Theorem 10) , we
have by Corollary 14.
Corollary 15. If f : X • Y is weakly-coη tinuous， 9 X • Y is gcontinuous , and Y is HausdorjJ, then {x E X : f(x) = g(x)} is g-closed.
209
On g-closed sets and g-continuous mapping
Finally, the fact that contiημ ity implies weak-continuity gives that Theorem A is a conseqμeηce of Corollary 15, 텐 X
is T 1 / 2 space
([8): A
topo-
logical space is T 1 / 2 iff eve 대 g-closed set is closed).
References
[1)
K. Balachandran , P . Sundaram and H. Maki , On Genernlized Continuo 따 Maps
in Topological Spaces , Mem. Fac. Sci. Kochi Univ . 12(1991) , 5- 13.
[2)
M. Caldas , Co η cerning Semi-Open and Semi-Closed Mappings , Sem. Bras. Analise
(SBA) , 35(1992) , 69-80
[3)
J. Dugundji , Topology , Allyn and Bacon , Boston 1966
[4)
W. Dunham and N . Levine , Further Resu lts on Genernlized Closed Sets in Topologi cal, Kyungpook Math . J . 20(1980) , 169-175
[5)
W . Dunham , A New Closure Operator for Non-T1 Topologies , Kyungpook Math.
J. 22(1982) , 55-60
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야)
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10 9- 11 1.
[7)
N. Levine , A Decomposition of Continuity in Topological Spaces , Amer.Math .
Monthly, 68(1961) , 44-46.
[8)
N. Levine , Genernlized Closed Sets in Topology , Rend. Cir. Mat . Palermo .
19(1970) , 89-9.6
[9)
T . Noiri , Between Continuity and
Weak-Co η tinu 때， Bol l. U.M. 1. 9(1974) , 647-654.
[10) T. Noiri , A Note on Semi-Continuous Mappings , Lincei- Re nd . Sc. Fisc. Mat. e
Nat . 55(1973) , 400-403 .
DEPARTAMENTO DE MATEMATICA APLICADA-IMUFF , UNIVERSIDADE FEDERAL FLUMINEl SE RUA São PAULO S/N , CEP: 24210 , NITEROI , RJ-BRASIL
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